Trilinear characterizations of the Fourier extension conjecture on the paraboloid in three dimensions

Abstract

We prove that a local trilinear extension inequality on the paraboloid in three dimensions is equivalent to the Fourier restriction conjecture, and then we prove a variant involving smooth Alpert wavelets that represents the weakest such inequality the authors could find that characterizes the Fourier extension conjecture.

21 pages, Theorem 10 has been slightly weakened by including small scales s_1 in Definition 9, due to an error in Case 3 that was discovered in the preparation of arXiv:2512.24990v5. All other results unchanged